During the last 20 years there has been a lot of progress in the design of machine learning models; however, their employment in scientific discovery of hidden physical laws of complex system is relatively recent. One of the most classical examples has been learning the material constitutive law directly from experimental data, where the governing partial differential equation (PDE) is unknown and the data features noises from measurements. This task can be formulated as identifying an operator that continuously maps the displacement field to the loading field. Moreover, such a need for learning operators between function spaces has also become vital in other applications ranging from homogenization problems [1]}, [2]}, [3]}, [4]}, fast PDE solvers [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, to control problems [6]}, [12]}, just to name a few. Among these work, models with nonlocal operators have
have received increasing attention, since they can describe physical phenomena that classical differential operators fail to capture and provide a powerful model for a large class of complex engineering and scientific applications [10]}, [14]}, [15]}, [16]}. However, despite a long line of work on nonlocal model learning and its applications,
there is limited theoretical characterization of the underlying inverse problem, even in the linear setting.
